Chapter 5 - The Six-Piece Burr

Puzzles consisting of interlocking assemblies of notched sticks are often
referred to as burr puzzles, probably from being pointed or spur-like in
assembled appearance. By far the most familiar of all burr puzzles, and probably
of three-dimensional puzzles in general, is the so-called six-piece burr. Its
origin is unknown, but it has been traced back to at least 1803 in Germany,
where it appears in a catalogue of G. H. Bestelmeier. Some persons know it as
the Chinese Puzzle or Chinese Cross, probably because it has been
mass-produced in the Orient since the early 1900s, but there does not appear to
be any evidence that the idea originated there.

David Bruce has put forth the plausible conjecture that some of the earliest
puzzles may have been but slight modifications of practical objects. For
example, note the familiar interlocking box shown in Fig. 63, consisting of six
notched boards. Did some whimsical box-maker decide to have fun one day in his
spare time, or did he perhaps just run out of nails? Whatever, this may well
have been the origin of the six-piece burr. With six identical pieces, as
suggested by the illustration, it is clearly impossible to assemble. There are
several obvious ways to modify one or more of the pieces to make it assemblable,
and a good exercise for the amateur puzzle-maker is to see how many of these
ways he or she can discover. With a penny slot, it becomes a toy bank - a good
first puzzle for any youngster.

Fig. 63

General Discussion

The standard six-piece burr consists of six notched square sticks of
arbitrary equal length not less than three times their width, arranged
symmetrically in three mutually perpendicular intersecting pairs. If the square
cross-section of the sticks has a dimension of two units, then all notches are
one unit deep and one unit wide or some exact multiple. To put it another way,
all notches can be regarded as being made by removal of discrete cubic units, or
to put it still another way, all pieces can be regarded as being built of cubic
units. All of the notches are made within the region of intersection with the
other sticks, so that when the puzzle is assembled no notches show and it has
apparent symmetry (Fig. 64).

Fig. 64

The six-piece burr is actually a large family of designs, since the designer
has a wide choice of how to notch each of the pieces. Over the years, variations
of the six-piece burr have received much attention from puzzle inventors and
authors. Directions for making them can be found in many books and magazines.
Several different versions have been manufactured and patented. The earliest US
patent is No.1,225,760 of Brown, dated 1917, with several others following
shortly thereafter. Most toy and novelty stores have a few burr puzzles on their
shelves or in their catalogues. Traditionally, these have been uninspired
time-worn versions with sliding key and internal symmetries. Consequently this
fine puzzle has suffered a chronically tarnished image. To make matters worse,
over the years many inventors have tinkered with bizarre embellishments to give
the basic burr puzzle their own stamp of identity. The patent files reveal many
such ill-conceived contraptions, including those with strings and holes, hidden
pins, rotating keys, and other secret locking devices. Evidently taking their
cue from certain composers of modern "music", they have thrown in odd
intervals, incongruously sharpened or flattened pieces, confusingly large
numbers of parts in hopeless disharmony with each other, and other jarring
complications. Within the last decade or two, though, the six-piece burr has
emerged from this decadent period to become once again the quintessential
interlocking puzzle, thanks largely to the work of Bill
Cutler.

There are twelve cubic units in each piece that are candidates for removal
(Fig. 65). Removing these in all possible permutations would theoretically
result in 4,096 different pieces, but by discarding symmetries and those that
cut the pieces in two, the number of practical pieces is 837. No one
knows how many different ways sets of six such pieces can be combined and
assembled, but they number in the millions. Cutler limited his analysis to only
those combinations that make a solid assembly with no internal voids. Using a
computer, he found there are 369 usable pieces and they can be assembled into a
solid burr 119,979 different ways. These results were published in the Journal
of Recreational Mathematics, Vol. 10(4), 1977-78, and were summarized
by Martin Gardner in Scientific
American, Mathematical Games column, Jan. 1978.

Editor's Note: Bill Cutler continued his
analysis of six-piece burrs culminating in the 1994 publication of A
Computer Analysis of All 6-Piece Burrs. His analysis showed that there
are roughly 35.65 billion possible assemblies (71.3 billion if mirror images are
counted). Of these, 35.65 billion assemblies, 5.95 billion can be taken apart.
The highest level found for a non-unique (more than one assembly) six-piece burr
was 12. The highest level unique six-piece burr is 10 if the pieces are 8 units
long and 9 if the pieces are 6 units long. If all pieces are notchable, the
highest level is 5 for a unique burr. His analysis completely explored all
assemblies for the first piece to be taken out. Only higher level burrs were
completely analyzed. So the number of total solvable burrs is a statistical
estimate.

Fig. 65

The burr pieces can be divided into two groups - those with simple notches
that can be milled out directly with a saw or dado blade (notchable
pieces), and those with blind corners and edges that must be chiseled out or
made by gluing in cubic blocks (unnotchable pieces). The notchable pieces
are the more desirable from both practical and aesthetic considerations. Some
puzzle analysts have limited their investigations to notchable pieces, of which
there are 59 (including the one with no notches). It has been customary to
consider only solid assemblies, in which case there are only 25 usable notchable
pieces, and these are commonly referred to as the set of 25 notchable pieces.
They can be chosen in sets of six and assembled solid in 314 different ways.
Some of this was calculated independently by several different analysts, with or
without a computer. All of it has been confirmed and organized by Cutler.

In Fig. 66, piece A is notchable. Piece B is not notchable.
Piece C can be made with a saw but cannot be assembled with other
notchable pieces without producing voids, so it is not included in the set of 25
notchable pieces, which are shown in Fig. 67.

Fig. 66

Fig 67

Burr No. 305

Cutler's computer analysis told only what was possible, not what was most
interesting. Actually it might possibly have done that too if appropriately
instructed. For example, from the list of the 314 solid notchable combinations,
suppose one first eliminates all those using duplicates (or triplicates) of
identical pieces and pieces having an axis of symmetry. Also eliminate
combinations with more than one solution. This narrows the field down to 18, of
which all but one (and its mirror image) employ a rather common and
uninteresting two-piece key arrangement. What emerges from this screening
process is a marvelous burr. It is called Burr No. 305 because of its
location in Cutler's tabulation. It uses pieces 6, 12, 14, 21, 22, and 23 (Fig.
68).

Fig. 68

Difficulty Index and Burr No. 306

This is an appropriate point at which to digress for a moment and introduce
the idea of a difficulty index for a combinatorial puzzle. Puzzles must
by definition have some element of difficulty. Making a puzzle more difficult
may in some circumstances be an improvement in design, if not carried to
extremes and if not to the detriment of other considerations. In any case, some
way of predicting the relative difficulty of similar puzzle designs would be a
useful tool for the designer.

Consider the solid six-piece burr. Given a drawing of the assembled burr or
some familiarity with it, the only real problem is determining the relative
location and orientation of the six pieces. Select any one of the six pieces at
random for the bottom piece. Usually it is obvious from the notching which side
should face the center. Now for the back piece, one has a choice of any of the
remaining five, and it can be turned end-for-end, hence a total of ten
possibilities. For the next piece, say on the left, there are six choices, and
so on. Thus, to make a complete analysis of the puzzle by trying every piece in
every position. There are a total of 10 x 8 x 6 x 4 x 2 or 3,840 possibilities
to be considered. This number divided by the number of solutions is the difficulty
index of that particular design.

The difficulty index of Burr No. 305 is 3,840. While that may seem like a
large number of moves, most of them are skipped by using common sense, and so
this would be a puzzle of medium difficulty. Identical pairs of pieces,
symmetrical pieces, and multiple solutions all decrease the difficulty index.
There is one charming type of piece known as an ambiguous piece, because
you cannot tell from the notches which side should face the center, and there
are different degrees of ambiguity. Piece No. 9 in Fig.
67 is an example of the most ambiguous type because any one of its four
sides might face the center. This would increase the difficulty index by an
additional factor of four, but because it is also symmetrical the net increase
would be a factor of two.

The mischievous role of the ambiguous piece was not taken into account in the
analysis that led to the illumination of Burr No. 305. Adding this newfound
ingredient to the recipe another delectable puzzle comes to light - Burr No. 306
illustrated in Fig. 69. It uses pieces 6, 9, 12, 21, 22, and 23, and has a
difficulty index of 7,680.

Note that a set of seven pieces will allow both Burr No. 305 and Burr No. 306
to be constructed.

Fig. 69

Higher-Level Burrs and Bill's Baffling Burr

So far we have discussed only burrs with no internal voids. Historically,
solid burrs have received the most attention. No satisfactory explanation has
ever been given for this, but perhaps it is simply the notion that many things
in life tend to be more satisfying when they are solid - building foundations,
financial investments, friendships, and so on. Also, Bill
Cutler points out that only by limiting his program to solid burrs was the
analysis practical as otherwise the computation time would have been too long. (Editor's
Note: This is no longer true. See the previous
note on this subject.) The recent flurry of activity in designing ever more
entertaining (meaning, to some, fiendishly difficult) burrs has shifted
attention to burrs that do not come directly apart (or go directly together) but
rather involve the shifting back and forth of pieces or groups of pieces within
the partially assembled burr. Some of these are so baffling as to discourage a
professional locksmith yet they are basically just standard burrs using the 837
practical pieces. They all necessarily have one or more internal voids.

One of the best of this new breed of burrs is Bill's Baffling Burr, designed
of course by Cutler. It uses two unnotchable pieces, both of which are easily
made from notchable pieces by gluing in one and two extra blocks. It has seven
internal voids. This is an unusually large number of voids for a burr with only
one solution, and contributes to its difficulty, for there are 24 apparent
solutions but only one that is possible to assemble. Thus, you may think you
have found the solution and are wondering how to get the last piece in place
when most likely you have stumbled upon one of the 23 false solutions. It was
stated earlier that the pieces could be of arbitrary length. With some of these
more complicated burrs, this is no longer true. Bill's Baffling Burr (Fig. 70)
cannot be assembled if the pieces are longer than three times their width.

Fig. 70

Bill's Baffling Burr is referred to as a level-five burr, meaning that
five separate shifts are required to release the first piece. This new yardstick
of devilry has spurred some rivalry among puzzle experts to see who can come up
with even higher level burrs. Around 1985, Gaby Games of Israel came out with an
amazing level-seven-four burr, meaning that seven moves are required to release
the first piece and four more are required to release the second piece. Would
someone next discover a level-eight burr? Well, not exactly. Recently Peter
Marineau surprised the puzzle world with a level-nine burr! Is this the upper
limit? Probably not. The Marineau burr, shown in Fig. 71 achieves its remarkable
stunt with surprisingly simple pieces. Two are identical, another two are a
reflexive pair, and another one is self-reflexive.

Fig. 71

Perhaps the reader will now be encouraged to wander off into this vast
wilderness of hidden notches and explore some of them further. For the puzzle
connoisseur, a well-crafted six-piece burr is the embodiment of good design -
simple, direct, and eminently functional. For the hobbyist, the burr is well
suited for a workshop project and helpful woodworking tips
are given later. In particular, the would-be puzzle inventor will find much to
explore beneath the deceptively familiar exterior of the six-piece burr.

Considering the large number of possible assemblable sets of the 837
practical pieces, recently estimated by Cutler to almost certainly exceed one
billion, any one of them chosen at random is likely to be a new and original,
but totally uninspired, design. The first step, then, is to decide just what
features one considers most desirable. A few guide-lines have been suggested
here, but there may be other, better ideas that have been overlooked.
Originality, psychology, and aesthetics all play a role at this stage of the
creative process. The second step is seeking the combination that best achieves
one's goal, and this is essentially an analytical and mechanical problem.

Imagine a computer being programmed to methodically print out all of the
probably billions of assemblable standard six-piece burrs. All but a handful
would be new and original designs - or would they? Does merely being different
constitute originality? There is a curious musical analogy. With conventional
discrete musical notation, one could, in theory at least, program a computer to
print out every possible musical theme, given enough time and unlimited supply
of paper. Buried within this mountain of papers would be all of the most sublime
works of the great masters of the past and of those perhaps to come in the next
Renaissance. But then how could they be found from amongst the random noise? The
whole exercise would amount to nothing.

Trying to improve upon an existing burr design can be an enlightening
exercise. For example, as a maker of puzzles, one is always trying to reduce the
number of unnotchable pieces. Moving or removing just one offending unit block
seems innocent enough, but it nearly always causes havoc. Attempts to correct
the problem just create more problems. (Sounds familiar?) Sometimes you work
through a loop of changes and end up back where you started. It is slow work for
every new change requires an analysis of all possible solutions. Some analysts
use a computer for this. It does in seconds what otherwise might take hours or
even days. Others not in such a rush may enjoy the mental exercise in
traditional methods of analysis using pencil and paper. For them, the analysis
is the puzzle so why not relax and enjoy it?